There's also a series formula, apparently (with respect to "w" in , so setting w = 1 yields the tetrational), but I haven't yet figured out how one is supposed to evaluate the general coefficient. How can that be done?

To test the analytic continuation, we can use the Cauchy integral: if is analytic, then we derive the powerseries coefficients at via

and

.

This seems to provide a more efficient algorithm for the recovery of the coefficients, than straight numerical differentiation from the difference quotient (which seems to require more rapidly-escalating levels of numerical precision).

We can now choose some close to , set for some fractional tower where is obtained from the regular formula, and a path that encircles it, but does not leave the kidneybean ("Shell-Thron" region) of convergence, e.g. a small circle round the point. Then, by increasing n, we obtain the Taylor coefficients. For , expanded about , using a circle of radius 0.01, we get the following estimates for the first 25 coefficients:

For , we can use this get , which is real, not complex. How does that agree with other methods of tetration for bases greater than ? This series should have radius of convergence 0.42, determined by the distance to the nearest singularity/branchpoint, which is at z = 1.

I'm not sure of a formal proof of the "continuability", though one approach may be to try and differentiate the regular iteration formula, then prove that the limit of the derivative as converges -- in order for it to switch to non-real complex values as is passed, that point would have to be some sort of singularity, like a branch point, and so the function would not be differentiable there, and if it is, then that is not the case.

I'll see if maybe I can get some graphs on the complex plane but calculating the regular iteration is a bear as it requires lots of numerical precision, at least for the limit formula. Maybe that series formula would be better?